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By definition, modified duration is
$$ D_\text{mod} = \frac{1}{P} \frac{dP}{dy} $$
where $P$ is the dirty price of a bond.
Clean price is the standard quoting convention for the vast majority of bond markets (though not all), but nearly all analytics, be it yield to maturity, DV01, or duration, are all computed using dirty price.

The specification of $ln(S)$ is based on the explicit assumption security prices and interest rates cannot go below zero.
And for the behaviour of securities, it has been well-established via empirical research that the security absolute price grows at an exponential rate rather than absolute rate.... i.e. after $T$ years, security price tends to be $S(0)e^{...

Intuitively, if there is 10x more depth at the bid than the ask, and trades arrive at (and are sized at) random, it is 10x more likely that the price will tick up than down.
In practice, often trades do not arrive at random and in fact show strong directional auto-correlation. Moreover, the depth at the bid/ask are not static quantities, they are constantly ...